Anomalous diffusion — particles spreading faster or slower than Brownian motion predicts — appears in dozens of complex systems: plasma turbulence, granular media, biological cell migration, financial markets. The standard mathematical description is the nonlinear Fokker-Planck equation, which modifies the classical diffusion framework by making the diffusion coefficient depend on the probability density itself. The nonlinearity is treated as a physical fact about the system — the medium is complicated, so the equation must be complicated.
Suyari shows that the nonlinearity is a coordinate artifact.
Starting from a simple growth law — dy/dx = y^q — he identifies the q-logarithm as the natural coordinate system for the problem. In these coordinates, the drift term is linear in the probability density, the Einstein relation (connecting diffusion to temperature) holds in its standard form, and the system obeys a conventional H-theorem. The stationary state is a q-Gaussian distribution that minimizes a free energy functional. All the machinery of equilibrium thermodynamics applies without modification.
The nonlinear Fokker-Planck equation is not a departure from classical physics. It is classical physics viewed through the wrong coordinate system. The q-logarithmic coordinates are the “natural” ones in a precise geometric sense: the state space of the anomalous system has a curved geometry, and the nonlinearity arises from forcing this curved space into Cartesian coordinates. Switch to coordinates that respect the curvature, and the equation becomes linear.
This is a stronger claim than “we found a useful mathematical transformation.” Suyari demonstrates a duality between the dynamic index q and the thermodynamic index 2−q: the same system has two equivalent descriptions, one dynamical and one thermodynamic, related by a specific algebraic transformation. The duality is not approximate — it is exact.
The general principle: what looks nonlinear from here may look linear from there. Complexity can be a property of the description, not the phenomenon. The decision of which coordinate system to use is not a notational convenience — it is a load-bearing choice that determines whether the system appears simple or complicated. Before adding complexity to the model, check whether the coordinates are creating the complexity you're trying to explain.